NUMERICAL METHODS FOR FINDING PERIODIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH STRONG NONLINEARITY

The paper introduces an ordinary differential equation consisting of a positively homogeneous main part of an order of m > 1 and a periodic perturbation. Approximate method of finding periodic solutions, which attempts to find zeros of explicitly defined finite-dimensional mappings, is presented.


INTRODUCTION
It is known that finding the exact solutions of a system of nonlinear differential equations is only possible in some particular cases. This problem becomes even more complicated when considering the issue of finding periodic solutions. It justifies the need for resorting in order to 6911 PERIODIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS approximate methods for finding periodic solutions, which in return, can provide important information around the phenomenon under study.
A considerable number of published articles is devoted to the problem of determining the conditions for finding the existence of periodic solutions that are related to differential equations and numerical methods. The existing conditions for periodic solutions are mainly related to the existence of a fixed point of some mappings, which are induced by a given equation, or by the application of variational methods in which the periodic solution is considered an extremal of some functions. In general, despite the fact that the proven fact of the periodic solution existence does not allow one to determine the points through which this solution passes, the problem of constructing numerical methods for finding a periodic solution is extremely significant.
It is known that  -periodicity in t of the right-hand side of the differential equation ( , ), does not imply the existence of a  -periodic solution. For example, a system of linear differential equations may even not contain  -periodic solutions, such as the linear Hamiltonian system, which is derived from [2]. A simpler example of a similar system can be provided as follows: It is obvious that this system does not have 2 -periodic solutions, in spite of the fact that 2 exists in the equation ( Thus, the problem of finding periodic solutions of a differential equation by numerical methods must be accompanied by an evidence for the existence of such solutions.
One of the numerical methods for finding periodic solutions is the Collocation method. The description of this method and its applications can be found in [2]. In some cases, periodic solutions can be found for a system of quasi-linear equations based on the use of the method that is proposed by A.M. Samoilenko [12]. A description of this method and references can be found in [3].
The purpose of this paper is to explore the periodic solutions of a differential equation with a positively homogeneous main part of an order of > 1 and with a periodic perturbation as: , : Such system as (1) is widely used in physics and in mathematical models that are applied in many different fields as engineering, chemistry, economy, finance, etc. [4].
System (1) was researched in a number of different previous studies. The existence of periodic solutions for system (1) was studied in [5].
Conditions for the existence of aprioristic estimate of periodic solutions and sufficient conditions for the existence of periodic solutions are all presented in [6][7][8]. There has been a few numbers of studies that are related to the problem of creating approximate methods for finding periodic solutions. For instance, the application of the collocation method does not always yield to satisfactory results for system (1). The reason behind this is that: If the main part of system (1) is positively homogeneous, and its order is > 1, one can deal with strong nonlinearity where not all solutions of system (1) are non-locally extended.
Next we assume that the functions are continuously differentiable, and the following condition is met:

M Replace system (1) by a different equation, and study this discrete analogue of Equation
(1).

Divide the interval into
Approximate a derivative as follows: Construct a difference scheme for finding periodic solutions. The periodicity condition for the solution implies that the following condition is met: 0 n xx = .
Considering the periodicity condition, the following discrete problem can be obtained: where, then, multiplying C by X gives: It is normal to assume that the solution of equation (8) is an approximate value of the periodic solution of system (1). Possible methods for estimating the rate of convergence of solutions of equation (8) to the solution of system (1) can be found in [11]. The existence of solutions of equation (8) requires the fulfilment of other additional conditions. Equation (8) could also be obtained as follows: Assume that system (1) is integrated over 11 [ ; ] ii tt −+ interval. The following equation is obtained: ( , ( )) ( , ( )) ( , ( )) 22  ( , ( )) ( , ( )) 4 ( , ( )) ( , ( )) 3 other discrete problems can be derived as follows: Further, assume that B is a matrix whose form is similar to (7), (9) or (10).

Theorem 1.
Assume that the conditions (2)-(5) are met. Then, the solutions of equation (8) Passing to a limit at k → , ( ) 0 BP U  = will be obtained. Nonetheless, in this case, We have arrived at a contradiction.
It follows from Theorem 1 that on the sphere of sufficiently large radius, the mappings 2 E Q hBF  = − + Where, ...

 =+
From Theorem 1, the following statement is obtained: Assume that Equation (11) has the solution at 0  = and for any function ( , ) p t x which meets Conditions (3) and (4) Further, assume that is an odd number, then in this case, Assume that 0 P is a projection on 0 , E and 1 P is a projection on 1 . E Let 01 X X X =+, (8) can be rewritten as follows: Then, the following equation can be obtained: PERIODIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS If h is sufficiently a small number, then the operator Thus, the approximate finding of the periodic solution of Equation (1) can be replaced with the finding of the system solution of Equation (13).
One can find the solution from System (13) using the following algorithm: For instance, consider a system of equations: In accordance with the above notation, we have:   The problem of finding periodic solutions for Equation (1) can be similarly considered for the case of homogeneity order of (0;1) m . In this case, the problem also attempts to find the solution of System (13).

CONCLUSION
The paper introduced an effective algorithm for an approximate finding of periodic solutions of the equations with positively homogeneous main parts. The algorithm can be applied both to the equations with homogeneity order of > 1, and to the equations with homogeneity order of 0 < < 1. In this paper, sufficient conditions for the aprioristic assessment of the existence of periodic solutions', and the condition for existence of periodic solutions are formulated. The statement on homotopy invariance of periodic solutions' existence for Equation (8) is announced. It will allow defining necessary and sufficient conditions for the existence of periodic solutions.